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### How to find more solutions for an periodical equation with infinity solutions

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```Dear all,

I have a matrix defined as:
Matrix[{\[Alpha]_, \[Beta]_, \[Gamma]_}, \[Theta]_] := {{\[Alpha]^2 \
(1 - Cos[\[Theta]]) +
Cos[\[Theta]], \[Alpha] \[Beta] (1 -
Cos[\[Theta]]) - \[Gamma] Sin[\[Theta]], \[A lpha] \[Gamma] (1 -
Cos[\[Theta]]) + \[Beta] Sin[\[Theta]]}, {\[Alpha] \[Beta] (1 \
- Cos[\[Theta]]) + \[Gamma] Sin[\[Theta]], \[Beta]^2 (1 -
Cos[\[Theta]]) +
Cos[\[Theta]], \[Beta] \[Gamma] (1 -
Cos[\[Theta]]) - \[Alpha] Sin[\[Theta]]}, {\[Alpha] \[Gamma] \
(1 - Cos[\[Theta]]) - \[Beta] Sin[\[Theta]], \[Beta] \[Gamma] (1 -
Cos[\[Theta]]) + \[Alpha] Sin[\[Theta]], \[Gamma]^2 (1 -
Cos[\[Theta]]) + Cos[\[Theta]]}}

Solve[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]].{x, y, z}, \[Theta]]

I know this equation is periodical and has infinity solutions. So, Mathematica only gave me only one solution: {{\[Theta] -> 0}} and show me the message:

Solve::ifun: Inverse functions are being used by Solve, so some solutions
may not be found; use Reduce for complete solution information. >>

My question is, how could I see for example first 6 solution, because I
know the first 5 or 6 solutions should be different and then
repeat themselves periodically. What should I do to find the first 5 or 6
solutions?

Thank you very much!
```
 0
Reply hitphyopt (5) 11/15/2011 10:51:18 AM

```Using Reduce will tell you much more about possible solutions (probably
everything):

Reduce[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]] . {x, y, z}, \[Theta]]

(C[1] \[Element] Integers && (\[Theta] == 2*Pi*C[1] || (x == 0 && y ==
0 && \[Theta] == Pi + 2*Pi*C[1]))) || ((-Pi + \[Theta])/(2*Pi)
\[NotElement] Integers && x == 0 && y == 0)
Your equation to be solved for \[Theta] obviously is looking for
Eigenvalues == 1 of the Matrix with Alpha = 0 and Beta = 0.

This can be accomplished more directly in the follwing way

ev = Eigenvalues[Matrix[{0, 0, 1}, \[Theta]]]
{1, Cos[\[Theta]] - I*Sin[\[Theta]], Cos[\[Theta]] +
I*Sin[\[Theta]]}

Reduce[ev[[1]] == 1, \[Theta]]
True

Reduce[ev[[2]] == 1, \[Theta]]
C[1] \[Element] Integers && \[Theta] == 2*Pi*C[1]

Reduce[ev[[3]] == 1, \[Theta]]
C[1] \[Element] Integers && \[Theta] == 2*Pi*C[1]

PS: I don't see the 5 or 6 solutions you mentioned, but only 3.

Regards,
Wolfgang

"Gy Peng" <hitphyopt@gmail.com> schrieb im Newsbeitrag
news:j9tg76\$n8d\$1@smc.vnet.net...
> Dear all,
>
> I have a matrix defined as:
> Matrix[{\[Alpha]_, \[Beta]_, \[Gamma]_}, \[Theta]_] := {{\[Alpha]^2 \
> (1 - Cos[\[Theta]]) +
>    Cos[\[Theta]], \[Alpha] \[Beta] (1 -
>       Cos[\[Theta]]) - \[Gamma] Sin[\[Theta]], \[A lpha] \[Gamma]
> (1 -
>        Cos[\[Theta]]) + \[Beta] Sin[\[Theta]]}, {\[Alpha] \[Beta] (1
> \
> - Cos[\[Theta]]) + \[Gamma] Sin[\[Theta]], \[Beta]^2 (1 -
>       Cos[\[Theta]]) +
>    Cos[\[Theta]], \[Beta] \[Gamma] (1 -
>       Cos[\[Theta]]) - \[Alpha] Sin[\[Theta]]}, {\[Alpha] \[Gamma] \
> (1 - Cos[\[Theta]]) - \[Beta] Sin[\[Theta]], \[Beta] \[Gamma] (1 -
>       Cos[\[Theta]]) + \[Alpha] Sin[\[Theta]], \[Gamma]^2 (1 -
>       Cos[\[Theta]]) + Cos[\[Theta]]}}
>
> Solve[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]].{x, y, z}, \[Theta]]
>
> I know this equation is periodical and has infinity solutions. So,
> Mathematica only gave me only one solution: {{\[Theta] -> 0}} and
> show me the message:
>
> Solve::ifun: Inverse functions are being used by Solve, so some
> solutions
> may not be found; use Reduce for complete solution information. >>
>
> My question is, how could I see for example first 6 solution, because
> I
> know the first 5 or 6 solutions should be different and then
> repeat themselves periodically. What should I do to find the first 5
> or 6
> solutions?
>
> Thank you very much!

```
 0
Reply weh (183) 11/16/2011 9:54:18 AM

1 Replies
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5/21/2013 10:20:03 AM